Question

Let $\int 2^{\ln x}dx=\frac{\left(f\right(x){)}^{\lambda }}{\lambda }+C,$ then which of the following is/are true (where $C$ is constant of integration)

• A. $f\left(x\right)=\ln x$
• B. $\lambda =\ln 2$
• C. $\lambda =1+\ln 2$
• D. $f\left(x\right)=x$

Answer 1

Answer: (C), (D)

$f\left(x\right)=x$

$\int 2^{\ln x}dx=\int x^{\ln 2}dx$
As we know that :
$[a^{{\log }_{b}c}=c^{{\log }_{b}a}]$
So, $\int x^{\ln 2}dx=\frac{x^{1+\ln 2}}{1+\ln 2}+C$